MODULE I HM

MODULE I

Conservation of Momentum: Momentum Theorem

In Newtonian mechanics, the conservation of momentum is defined by Newtons second law of

motion.

Newtons Second Law of Motion

The rate of change of momentum of a body is proportional to the impressed action and

takes place in the direction of the impressed action.

If a force acts on the body, linear momentum is implied.

If a torque (moment) acts on the body, angular momentum is implied.

Application of momentum theorem

Application of momentum theorem in some practical cases of inertial and non-inertial control

volumes is presented below

Inertial Control Volumes

Applications of momentum theorem for an inertial control volume are described with reference

to three distinct types of practical problems, namely

Forces acting due to internal flows through expanding or reducing pipe bends.

Forces on stationary and moving vanes due to impingement of fluid jets.

Jet propulsion of ship and aircraft moving with uniform velocity.

Non-inertial Control Volume

A good example of non-inertial control volume is a rocket engine which works on the principle

of jet propulsion.

Forces due to Flow Through Expanding or Reducing Pipe Bends

Let us consider a fluid flow through an expander shown in Fig. 1.1a below. The expander is held

in a vertical plane. The inlet and outlet velocities are given by V1 and V2 as shown in the figure.

The inlet and outlet pressures are also prescribed as p1 and p2. The velocity and pressure at inlet

and at outlet sections are assumed to be uniform. The problem is usually posed for the estimation

of the force required at the expander support to hold it in position.

Fig 1.1a Flow of a fluid through an expander

For the solution of this type of problem, a control volume is chosen to coincide with the interior

of the expander as shown in Fig. 1.1a. The control volume being constituted by areas 1-2, 2-3, 3-

4, and 4-1 is shown separately in Fig.1.1b.

The external forces on the fluid over areas 2-3 and 1-4 arise due to net efflux of linear

momentum through the interior surface of the expander. Let these forces be Fx and Fy. Since the

control volume 1234 is stationary and at a steady state, we have for x and y components

(1.1a)

and

(1.1b)

or,

(1.2a)

and

(1.2b)

where = mass flow rate through the expander. Analytically it can be expressed as

where A1 and A2 are the cross-sectional areas at inlet and outlet of the expander and the flow is

considered to be incompressible.

M represents the mass of fluid contained in the expander at any instant and can be expressed as

where is the internal volume of the expander.

Thus, the forces Fx and Fy acting on the control volume (Fig. 1.1b) are exerted by the expander.

According to Newtons third law, the expander will experience the forces Rx (= Fx) and Ry ( =

Fy) in the x and y directions respectively as shown in the free body diagram of the expander. in

fig 1.1c.

Fig 1.1b Control Volume Comprising the fluid

contained in the expander at any instant

Fig 1.1c Free Body Diagram of the

Expander

The expander will also experience the atmospheric pressure force on its outer surface. This is

shown separately in Fig. 1.2.

Fig 1.2 Effect of atmospheric pressure on the expander

From Fig.1.2 the net x and y components of the atmospheric pressure force on the expander can

be written as

The net force on the expander is therefore,

(1.3a)

(1.3b)

or,

(1.4a)

(1.4b)

Note: At this stage that if Fx and Fy are calculated from the Eqs (1.2a) and (1.2b) with p1 and p2

as the gauge pressures instead of the absolute ones the net forces on the expander Ex and Ey will

respectively be equal to Fx and Fy.

Dynamic Forces on Plane Surfaces due to the Impingement of Liquid Jets

Force on a stationary surface Consider a stationary flat plate and a liquid jet of cross sectional

area a striking with a velocity V at an angle to the plate as shown in Fig. 1.3a.

Fig 1.3 Impingement of liquid Jets on a Stationary Flat Plate

To calculate the force required to keep the plate stationary, a control volume ABCDEFA (Fig.

1.3a) is chosen so that the control surface DE coincides with the surface of the plate. The control

volume is shown separately as a free body in Fig. 1.3b. Let the volume flow rate of the incoming

jet be Q and be divided into Q1 and Q2 gliding along the surface (Fig. 1.3a) with the same

velocity V since the pressure throughout is same as the atmospheric pressure, the plate is

considered to be frictionless and the influence of a gravity is neglected (i.e. the elevation between

sections CD and EF is negligible).

Coordinate axes are chosen as 0s and 0n along and perpendicular to the plate respectively.

Neglecting the viscous forces. (the force along the plate to be zero),the momentum conservation

of the control volume ABCDEFA in terms of s and n components can be written as

(1.5a)

and

(1.5b)

where Fs and Fn are the forces acting on the control volume along 0s and 0n respectively,

From continuity,

Q = Q1 + Q2 (1.6)

With the help of Eqs (1.5a) and (1.6), we can write

(1.7a)

(1.7b)

The net force acting on the control volume due to the change in momentum of the jet by the plate

is Fn along the direction "On and is given by the Eq. (1.7b) as

(1.7c)

Hence, according to Newtons third law, the force acting on the plate is

(1.8)

If the cross-sectional area of the jet is a, then the volume flow rate Q striking the plate can be

written as Q = aV. Equation (1.8) then becomes

(1.9)

Stationary vane problem

Consider a jet that is deflected by a stationary vane, such as is given in Fig. 1.4. If the jet speed

and diameter are 25 m/s and 25 cm, respectively and jet is deflected 600, what force is exerted by

the jet on the vane?

Fig 1.4

First solve for Fx , the x-component of force of the vane on the jet -

Here, the final velocity in the x-direction is given as

Hence,

also,

and

Therefore,

similarly determined, the y-component of force on the jet is

Then the force on the vane will be the reactions to the forces of the vane on the jet, or

Force on a moving surface

Fig 1.5 Impingement of liquid jet on a moving flat plate

If the plate in the above problem moves with a uniform velocity u in the direction of jet velocity

V (Fig. 1.5). The volume of the liquid striking the plate per unit time will be

Q = a(V u) (1.10)

Physically, when the plate recedes away from the jet it receives a less quantity of liquid per unit

time than the actual mass flow rate of liquid delivered, say by any nozzle. When u = V, Q = 0

and when u > V, Q becomes negative. This implies physically that when the plate moves away

from the jet with a velocity being equal to or greater than that of the jet, the jet can never strike

the plate.

The control volume ABCDEFA in the case has to move with the velocity u of the plate. We have

to calculate the forces acting on the control volume. Hence the velocities relative to the control

volume will come into picture. The velocity of jet relative to the control volume at its inlet

becomes VR1 = V u

Since the pressure remains same throughout, the magnitudes of the relative velocities of liquid at

outlets become equal to that at inlet, provided the friction between the plate and the liquid is

neglected. Moreover, for a smooth shockless flow, the liquid has to glide along the plate and

hence the direction of VR0, the relative velocity of the liquid at the outlets, will be along the

plate. The absolute velocities of the liquid at the outlets can be found out by adding vectorially

the plate velocity u and the relative velocity of the jet V - u with respect to the plate. This is

shown by the velocity triangles at the outlets (Fig. 1.5). Coordinate axes fixed to the control

volume ABCDEFA are chosen as 0s and 0n along and perpendicular to the plate

respectively.

The force acting on the control volume along the direction 0s will be zero for a frictionless

flow. The net force acting on the control volume will be along 0n only. To calculate this force

Fn, the momentum theorem with respect to the control volume ABCDEFA can be written as

Substituting Q from Eq (1.10),

Hence the force acting on the plate becomes

(1.11)

If the plate moves with a velocity u in a direction opposite to that of V (plate moving towards the

jet), the volume of liquid striking the plate per unit time will be Q = a(V + u) and, finally, the

force acting on the plate would be

(1.12)

From the comparison of the Eq. (1.9) with Eqs (1.11) and (1.12), conclusion can be drawn that

for a given value of jet velocity V, the force exerted on a moving plate by the jet is either greater

or lower than that exerted on a stationary plate depending upon whether the plate moves towards

the jet or away from it respectively. The power developed due to the motion of the plate can be

written (in case of the plate moving in the same direction as that of the jet) as

P = Fp . U

(1.13)

Dynamic Forces on Curve Surfaces due to the Impingement of Liquid Jets

The principle of fluid machines is based on the utilization of useful work due to the force exerted

by a fluid jet striking and moving over a series of curved vanes in the periphery of a wheel

rotating about its axis. The force analysis on a moving curved vane is understood clearly from

the study of the inlet and outlet velocity triangles as shown in Fig. 1.6.

The fluid jet with an absolute velocity V1 strikes the blade at the inlet. The relative velocity of

the jet Vr1 at the inlet is obtained by subtracting vectorially the velocity u of the vane from V1.

The jet strikes the blade without shock if 1 (Fig. 1.6) coincides with the inlet angle at the tip of

the blade. If friction is neglected and pressure remains constant, then the relative velocity at

the outlet is equal to that at the inlet (Vr2 = Vr1).

Fig 1.6 Flow of Fluid along a Moving Curved Plane

The control volume as shown in Fig. 1.6 is moving with a uniform velocity u of the vane.

Therefore we have to use Eq.(10.18d) as the momentum theorem of the control volume at its

steady state. Let Fc be the force applied on the control volume by the vane. Therefore we can

write

To keep the vane translating at uniform velocity, u in the direction as shown. the force F has to

act opposite to Fc Therefore,

(1.14)

From the outlet velocity triangle, it can be written

or,

or,

or,

(1.15a)

Similarly from the inlet velocity triangle. it is possible to write

(1.15b)

Addition of Eqs (1.15a) and (1.15b) gives

Power developed is given by

(1.16)

The efficiency of the vane in developing power is given by

(1.17)

Propulsion of a Ship

Jet propulsion of ship is found to be less efficient than propulsion by screw propeller due to the

large amount of frictional losses in the pipeline and the pump, and therefore, it is used rarely. Jet

propulsion may be of some advantage in propelling a ship in a very shallow water to avoid

damage of a propeller.

Consider a jet propelled ship, moving with a velocity V, scoops water at the bow and discharges

astern as a jet having a velocity Vr relative to the ship.The control volume is taken fixed to the

ship as shown in Fig. 1.7.

Fig 1.7 A control volume for a moving ship

Following the momentum theorem as applied to the control volume shown. We can write

Where Fc is the external force on the control volume in the direction of the ships motion. The

forward propulsive thrust F on the ship is given by

(1.18)

Propulsive power is given by

Application of Moment of Momentum Theorem

Let us take an example of a sprinkler like turbine . The turbine rotates in a horizontal plane with

angular velocity . The radius of the turbine is r. Water enters the turbine from a vertical pipe

that is coaxial with the axis of rotation and exits through the nozzles of cross sectional area a

with a velocity Ve relative to the nozzle.

A control volume with its surface around the turbine is also shown in the fig below.

Fig 2.1 A Sprinkler like Turbine

Application of Moment of Momentum Theorem (Eq. 10.20b) gives

(2.1)

When Mzc is the moment applied to the control volume. The mass flow rate of water through the

turbine is given by

The velocity must be referenced to an inertial frame so that

(2.2)

The moment Mz acting on the turbine can be written as

(2.3)

The power produced by the turbine is given by

(2.4)

Basic Principles of Turbomachines

A fluid machine is a device which converts the energy stored by a fluid into mechanical energy or vice versa. The energy stored by a fluid mass appears in the form of potential, kinetic

and intermolecular energy. The mechanical energy, on the other hand, is usually transmitted by a

rotating shaft. Machines using liquid (mainly water, for almost all practical purposes) are

termed as hydraulic machines.

CLASSIFICAITONS OF FLUID MACHINES

The fluid machines may be classified under different categories as follows:

(A) Classification Based on Direction of Energy Conversion

The device in which the kinetic, potential or intermolecular energy held by the fluid is converted

in the form of mechanical energy of a rotating member is known as a turbine.

The machines, on the other hand, where the mechanical energy from moving parts is transferred

to a fluid to increase its stored energy by increasing either its pressure or velocity are known as

pumps, compressors, fans or blowers.

(B) Classification Based on Principle of Operation

The machines whose functioning depend essentially on the change of volume of a certain amount

of fluid within the machine are known as positive displacement machines.

The word positive displacement comes from the fact that there is a physical displacement of the

boundary of a certain fluid mass as a closed system. This principle is utilized in practice by the

reciprocating motion of a piston within a cylinder while entrapping a certain amount of fluid in

it. Therefore, the word reciprocating is commonly used with the name of the machines of this

kind.

The machine producing mechanical energy is known as reciprocating engine while the machine

developing energy of the fluid from the mechanical energy is known as reciprocating pump or

reciprocating compressor.

The machines, functioning of which depend basically on the principle of fluid dynamics, are

known as rotodynamic machines . They are distinguished from positive displacement machines

in requiring relative motion between the fluid and the moving part of the machine. The rotating

element of the machine usually consisting of a number of vanes or blades, is known as rotor or

impeller while the fixed part is known as stator. Impeller is the heart of rotodynamic machines,

within which a change of angular momentum of fluid occurs imparting torque to the rotating

member.

For turbines, the work is done by the fluid on the rotor, while, in case of pump, compressor, fan

or blower, the work is done by the rotor on the fluid element.

Depending upon the main direction of fluid path in the rotor, the machine is termed as radial

flow or axial flow machine. In radial flow machine, the main direction of flow in the rotor is

radial while in axial flow machine, it is axial.

For radial flow turbines, the flow is towards the centre of the rotor, while, for pumps and

compressors, the flow is away from the centre. Therefore, radial flow turbines are sometimes

referred to as radially inward flow machines and radial flow pumps as radially outward flow

machines. Examples of such machines are the Francis turbines and the centrifugal pumps or

compressors. The examples of axial flow machines are Kaplan turbines and axial flow

compressors.

If the flow is party radial and partly axial, the term mixed-flow machine is used. Figure 1.1 (a)

(b) and (c) are the schematic diagrams of various types of impellers based on the flow direction.

Fig. 1.1 Schematic of different types of impellers

(C) Classification Based on Fluid Used

The fluid machines use either liquid or gas as the working fluid depending upon the purpose. The

machine transferring mechanical energy of rotor to the energy of fluid is termed as a pump when

it uses liquid, and is termed as a compressor or a fan or a blower, when it uses gas.

The compressor is a machine where the main objective is to increase the static pressure of a gas.

Therefore, the mechanical energy held by the fluid is mainly in the form of pressure energy.

Fans or blowers, on the other hand, mainly cause a high flow of gas, and hence utilize the

mechanical energy of the rotor to increase mostly the kinetic energy of the fluid. In these

machines, the change in static pressure is quite small.

For all practical purposes, liquid used by the turbines producing power is water, and therefore,

they are termed as water turbines or hydraulic turbines.

Turbines handling gases in practical fields are usually referred to as steam turbine, gas turbine,

and air turbine depending upon whether they use steam, gas (the mixture of air and products of

burnt fuel in air) or air.

ROTODYNAMIC MACHINES

The important element of a rotodynamic machine, in general, is a rotor consisting of a number of

vanes or blades. There always exists a relative motion between the rotor vanes and the fluid. The

fluid has a component of velocity and hence of momentum in a direction tangential to the rotor.

While flowing through the rotor, tangential velocity and hence the momentum changes.

The rate at which this tangential momentum changes corresponds to a tangential force on the

rotor. In a turbine, the tangential momentum of the fluid is reduced and therefore work is done

by the fluid to the moving rotor. But in case of pumps and compressors there is an increase in the

tangential momentum of the fluid and therefore work is absorbed by the fluid from the moving

rotor.

Basic Equation of Energy Transfer in Rotodynamic Machines

The basic equation of fluid dynamics relating to energy transfer is same for all rotodynamic

machines and is a simple form of Newtons Laws of Motion" applied to a fluid element

traversing a rotor. Here we shall make use of the momentum theorem as applicable to a fluid

element while flowing through fixed and moving vanes. a rotor of a generalised fluid machine,

with 0-0 the axis of rotation and the angular velocity. Fluid enters the rotor at 1, passes

through the rotor by any path and is discharged at 2. The points 1 and 2 are at radii and from

the centre of the rotor, and the directions of fluid velocities at 1 and 2 may be at any arbitrary

angles. For the analysis of energy transfer due to fluid flow in this situation, we assume the

following:

(a) The flow is steady, that is, the mass flow rate is constant across any section (no storage or

depletion of fluid mass in the rotor).

(b) The heat and work interactions between the rotor and its surroundings take place at a

constant rate.

(c) Velocity is uniform over any area normal to the flow. This means that the velocity vector at

any point is representative of the total flow over a finite area. This condition also implies that

there is no leakage loss and the entire fluid is undergoing the same process.

The velocity at any point may be resolved into three mutually perpendicular components as

shown in Fig 1.2. The axial component of velocity is directed parallel to the axis of rotation,

the radial component is directed radially through the axis to rotation, while the tangential

component is directed at right angles to the radial direction and along the tangent to the rotor

at that part.

The change in magnitude of the axial velocity components through the rotor causes a change in

the axial momentum. This change gives rise to an axial force, which must be taken by a thrust

bearing to the stationary rotor casing. The change in magnitude of radial velocity causes a

change in momentum in radial direction.

Fig 1.2 Components of flow velocity in a generalised fluid

machine

However, for an axisymmetric flow, this does not result in any net radial force on the rotor. In

case of a non uniform flow distribution over the periphery of the rotor in practice, a change in

momentum in radial direction may result in a net radial force which is carried as a journal load.

The tangential component only has an effect on the angular motion of the rotor. In

consideration of the entire fluid body within the rotor as a control volume, we can write from the

moment of momentum theorem

(1.1)

where T is the torque exerted by the rotor on the moving fluid, m is the mass flow rate of fluid

through the rotor. The subscripts 1 and 2 denote values at inlet and outlet of the rotor

respectively. The rate of energy transfer to the fluid is then given by

(1.2)

where is the angular velocity of the rotor and which represents the linear velocity of

the rotor. Therefore and are the linear velocities of the rotor at points 2 (outlet ) and 1

(inlet) respectively (Fig. 1.2). The Eq, (1.2) is known as Euler's equation in relation to fluid

machines. The Eq. (1.2) can be written in terms of head gained 'H' by the fluid as

(1.3)

In usual convention relating to fluid machines, the head delivered by the fluid to the rotor is

considered to be positive and vice-versa. Therefore, Eq. (1.3) written with a change in the sign of

the right hand side in accordance with the sign convention as

(1.4)

Components of Energy Transfer

It is worth mentioning in this context that either of the Eqs. (1.2) and (1.4) is applicable

regardless of changes in density or components of velocity in other directions. Moreover, the

shape of the path taken by the fluid in moving from inlet to outlet is of no consequence. The

expression involves only the inlet and outlet conditions. A rotor, the moving part of a fluid

machine, usually consists of a number of vanes or blades mounted on a circular disc. Figure 1.3a

shows the velocity triangles at the inlet and outlet of a rotor. The inlet and outlet portions of a

rotor vane are only shown as a representative of the whole rotor.

(a) (b)

Fig 1.3 (a) Velocity triangles for a generalised rotor vane

Fig 1.3 (b) Centrifugal effect in a flow of fluid with rotation

Vector diagrams of velocities at inlet and outlet correspond to two velocity triangles, where is

the velocity of fluid relative to the rotor and are the angles made by the directions of the

absolute velocities at the inlet and outlet respectively with the tangential direction, while and

are the angles made by the relative velocities with the tangential direction. The angles and

should match with vane or blade angles at inlet and outlet respectively for a smooth,

shockless entry and exit of the fluid to avoid undesirable losses. Now we shall apply a simple

geometrical relation as follows:

From the inlet velocity triangle,

or, (1.5)

Similarly, from the outlet velocity triangle,

or, (1.6)

Invoking the expressions of and in Eq. (1.4), we get H (Work head, i.e. energy per

unit weight of fluid, transferred between the fluid and the rotor as) as

(1.7)

The Eq (1.7) is an important form of the Euler's equation relating to fluid machines since it

gives the three distinct components of energy transfer as shown by the pair of terms in the round

brackets. These components throw light on the nature of the energy transfer. The first term of Eq.

(1.7) is readily seen to be the change in absolute kinetic energy or dynamic head of the fluid

while flowing through the rotor. The second term of Eq. (1.7) represents a change in fluid energy

due to the movement of the rotating fluid from one radius of rotation to another.

More about Energy Transfer in Turbomachines

Equation (1.7) can be better explained by demonstrating a steady flow through a container

having uniform angular velocity as shown in Fig.1.3b. The centrifugal force on an

infinitesimal body of a fluid of mass dm at radius r gives rise to a pressure differential dp across

the thickness dr of the body in a manner that a differential force of dp.dA acts on the body

radially inward. This force, in fact, is the centripetal force responsible for the rotation of the fluid

element and thus becomes equal to the centrifugal force under equilibrium conditions in the

radial direction. Therefore, we can write

with dm = dA dr where is the density of the fluid, it becomes

For a reversible flow (flow without friction) between two points, say, 1 and 2, the work done per

unit mass of the fluid (i.e., the flow work) can be written as

The work is, therefore, done on or by the fluid element due to its displacement from radius to

radius and hence becomes equal to the energy held or lost by it. Since the centrifugal force

field is responsible for this energy transfer, the corresponding head (energy per unit weight)

is termed as centrifugal head. The transfer of energy due to a change in centrifugal head

causes a change in the static head of the fluid.

The third term represents a change in the static head due to a change in fluid velocity relative to

the rotor. This is similar to what happens in case of a flow through a fixed duct of variable cross-

sectional area. Regarding the effect of flow area on fluid velocity relative to the rotor, a

converging passage in the direction of flow through the rotor increases the relative velocity

and hence decreases the static pressure. This usually happens in case of turbines.

Similarly, a diverging passage in the direction of flow through the rotor decreases the relative

velocity and increases the static pressure as occurs in case of pumps and

compressors.

Energy Transfer in Axial Flow Machines

For an axial flow machine, the main direction of flow is parallel to the axis of the rotor, and

hence the inlet and outlet points of the flow do not vary in their radial locations from the axis of

rotation. Therefore, and the equation of energy transfer Eq. (1.7) can be written, under

this situation, as

(2.2)

Hence, change in the static head in the rotor of an axial flow machine is only due to the flow of

fluid through the variable area passage in the rotor.

Radially Outward and Inward Flow Machines

For radially outward flow machines, , and hence the fluid gains in static head, while, for

a radially inward flow machine, and the fluid losses its static head. Therefore, in radial

flow pumps or compressors the flow is always directed radially outward, and in a radial flow

turbine it is directed radially inward.

Impulse and Reaction Machines

The relative proportion of energy transfer obtained by the change in static head and by the

change in dynamic head is one of the important factors for classifying fluid machines. The

machine for which the change in static head in the rotor is zero is known as impulse machine . In

these machines, the energy transfer in the rotor takes place only by the change in dynamic head

of the fluid.

The parameter characterizing the proportions of changes in the dynamic and static head in the

rotor of a fluid machine is known as degree of reaction and is defined as the ratio of energy

transfer by the change in static head to the total energy transfer in the rotor.

Therefore, the degree of reaction,

(2.3)

Impulse and Reaction Machines

For an impulse machine R = 0, because there is no change in static pressure in the rotor. It is

difficult to obtain a radial flow impulse machine, since the change in centrifugal head is obvious

there. Nevertheless, an impulse machine of radial flow type can be conceived by having a change

in static head in one direction contributed by the centrifugal effect and an equal change in the

other direction contributed by the change in relative velocity. However, this has not been

established in practice. Thus for an axial flow impulse machine, .

For an impulse machine, the rotor can be made open, that is, the velocity V1 can represent an

open jet of fluid flowing through the rotor, which needs no casing. A very simple example of an

impulse machine is a paddle wheel rotated by the impingement of water from a stationary nozzle

as shown in Fig.2.1a.

Fig 2.1 (a) Paddle wheel as an example of impulse turbine

(b) Lawn sprinkler as an example of reaction turbine

A machine with any degree of reaction must have an enclosed rotor so that the fluid cannot

expand freely in all direction. A simple example of a reaction machine can be shown by the

familiar lawn sprinkler, in which water comes out (Fig. 2.1b) at a high velocity from the rotor in

a tangential direction. The essential feature of the rotor is that water enters at high pressure and

this pressure energy is transformed into kinetic energy by a nozzle which is a part of the rotor

itself.

In the earlier example of impulse machine (Fig. 2.1a), the nozzle is stationary and its function is

only to transform pressure energy to kinetic energy and finally this kinetic energy is transferred

to the rotor by pure impulse action. The change in momentum of the fluid in the nozzle gives rise

to a reaction force but as the nozzle is held stationary, no energy is transferred by it. In the case

of lawn sprinkler (Fig. 2.1b), the nozzle, being a part of the rotor, is free to move and, in fact,

rotates due to the reaction force caused by the change in momentum of the fluid and hence the

word reaction machine follows.

Efficiencies

The concept of efficiency of any machine comes from the consideration of energy transfer and is

defined, in general, as the ratio of useful energy delivered to the energy supplied. Two

efficiencies are usually considered for fluid machines-- the hydraulic efficiency concerning the

energy transfer between the fluid and the rotor, and the overall efficiency concerning the energy

transfer between the fluid and the shaft. The difference between the two represents the energy

absorbed by bearings, glands, couplings, etc. or, in general, by pure mechanical effects which

occur between the rotor itself and the point of actual power input or output.

Therefore, for a pump or compressor,

(2.4a)

(2.4b)

For a turbine,

(2.5a)

(2.5b)

The ratio of rotor and shaft energy is represented by mechanical efficiency .

Therefore

(2.6)